RF ENGINEERING - BASIC CONCEPTS F. Caspers, P. McIntosh, T.

Kroyer ABSTRACT The concept of describing RF circuits in terms of waves is discussed and the S matrix and related matrices are defined. The signal flow graph (SFG) is introduced as a graphical means to visualise how waves propagate in an RF network. The properties of the most relevant passive RF devices (hybrids, couplers, non-reciprocal elements, etc.) are delineated and the corresponding S parameters are given. For microwave integrated circuits (MICs) planar transmission lines such as the microstrip line have become very important. A brief discussion on the Smith Chart concludes this paper. 1. INTRODUCTION

For the design of RF and microwave circuits a practical tool is required. The linear dimensions of the elements that are in use may be of the order of one wavelength or even larger. In this case the equivalent circuits which are commonly applied for lower frequencies lead to difficulties in the definition of voltages and currents. A description in terms of waves becomes more meaningful. These waves are scattered (reflected, transmitted) in RF networks. Having introduced certain definitions of the relation between voltages, currents and waves we discuss the S and T matrices such for the description of 2-port networks. Nowadays the calculation of complex microwave networks is usually carried out by means of computer codes. These apply matrix descriptions and conversions extensively. Another way to analyze microwave networks is by taking advantage of the signal flow graph (SFG). The SFG is a graphical representation of a system linear equations and permits one to visualise how, for example, an incident wave propagates through the network. However, for a systematic analysis of large networks the SFG is not very convenient; computer codes implementing the matrix formulation are generally used these days. In a subsequent section the properties of typical microwave n-ports (n = 1, 2, 3, 4) are discussed. The n-ports include power dividers, directional couplers, circulators and 180° hybrids. Historically many microwave elements have been built first in waveguide technology. Today waveguide technology is rather restricted to high-power applications or for extremely high frequencies. Other less bulky types of transmission lines have been developed such as striplines and micro striplines. They permit the realisation of microwave integrated circuits (MICs) or, if implemented on a semiconductor substrate, the monolithic microwave integrated circuits (MMICs). This paper concludes with a description of the Smith Chart, a graphical method of evaluating the complex reflection coefficient for a given load. Several examples including the coupling of single-cell resonators are mentioned. 2. S PARAMETERS

The abbreviation S has been derived from the word scattering. For high frequencies, it is convenient to describe a given network in terms of waves rather than voltages or currents. This permits an easier definition of reference planes. For practical reasons, the description in terms of in- and outgoing waves has been introduced. Now, a 4-pole network becomes a 2-port and a 2npole becomes an n-port. In the case of an odd pole number (e.g. 3-pole), a common reference point may be chosen, attributing one pole equally to two ports. Then a 3-pole is converted into a (3+1) pole corresponding to a 2-port. As a general conversion rule for an odd pole number one more pole is added.

1

I1

I2

Fig. 1 Example for a 2-port network: A series impedance Z. Let us start by considering a simple 2-port network consisting of a single impedance Z connected in series (Fig. 1). The generator and load impedances are ZG and ZL, respectively. If Z = 0 and ZL = ZG (for real ZG) we have a matched load, i.e. maximum available power goes into the load and U1 = U2 = U0/2. Please note that all the voltages and currents are peak values. The lines connecting the different elements are supposed to have zero electrical length. Connections with a finite electrical length are drawn as double lines or as heavy lines. Now we would like to relate U0, U1 and U2 with a and b. Definition of “power waves” The waves going towards the n-port are a = (a1, a2, ..., an), the waves travelling away from the n-port are b = (b1, b2, ..., bn). By definition currents going into the n-port are counted positively and currents flowing out of the n-port negatively. The wave a1 is going into the n-port at port 1 is derived from the voltage wave going into a matched load. In order to make the definitions consistent with the conservation of energy, the voltage is Z 0 . Z0 is in general an arbitrary reference impedance, but usually the normalized to characteristic impedance of a line (e.g. Z0 = 50 Ω) is used and very often ZG = ZL = Z0. In the following we assume Z0 to be real. The definitions of the waves a1 and b1 are

a1 =

U0 2 Z0

=

incident voltage wave ( port 1) Z0

U inc = 1 Z0
(2.1)

refl U1 reflected voltage wave ( port 1) b1 = = Z0 Z0

Note that a and b have the dimension

power [1].

The power travelling towards port 1, P1inc, is simply the available power from the source, while the power coming out of port 1, P1refl, is given by the reflected voltage wave.

2

inc P 1 =

1 2 a1 = = 2 2Z0
refl 1 2 U1 b1 = = 2 2Z 0

inc U1

2

inc I1

2

2

Z0
(2.2)

refl P = 1

refl 2 I1

2

Z0

Please note the factor 2 in the denominator, which comes from the definition of the voltages and currents as peak values (“European definition”). In the “US definition” effective values are used and the factor 2 is not present, so for power calculations it is important to check how the voltages are defined. For most applications, this difference does not play a role since ratios of waves are used. In the case of a mismatched load ZL there will be some power reflected towards the 2-port from ZL
inc P2 =

1 2 a2 2

(2.3)

There is also the outgoing wave of port 2 which may be considered as the superimposition of a wave that has gone through the 2-port from the generator and a reflected part from the mismatched load. We have defined a1 = U 0

(2

Z 0 = U inc Z 0 with the incident voltage wave

)

Uinc. In analogy to that we can also quote a1 = I inc Z 0 with the incident current wave Iinc. We obtain the general definition of the waves ai travelling into and bi travelling out of an n-port:

ai = bi =

Ui + Ii Z0 2 Z0 Ui − Ii Z0 2 Z0
(2.4)

Solving these two equations, Ui and Ii can be obtained for a given ai and bi as

parallel branches
xy 2.where M and M’ are square matrices with n rows and columns. When there are no direct signal loops. Each wave ai and bi is represented by a node. Examples We are looking for the input reflection coefficient b1/a1 of a two-port with a non-matched load ρL and a matched generator (source ρS = 0). For not too complicated circuits. Care has to be taken applying the third rule. Resolve loops
x x y y
x
y
x+y 1. For general problems the SFG can be solved for applying Mason’s rule (see Appendix II). Add the signal of parallel branches 2. Multiply the signals of cascaded branches 3. 4): 1. 5). each arrow stands for an S parameter (Fig. The elements of M and M’ appear as transmission coefficients of the signal path. 5.2)
The SFG can be drawn as a directed graph. as is generally the case in practise.
6
. the previous equation simplifies to y = Mx. No signal paths should be interrupted when resolving loops. loops
Fig. x represents the n independent variables (sources) and y the n dependent variables. which is equivalent to the usual S parameter definition b = Sa (3. a more intuitive way is to simplify step-by-step the SFG by applying the following three rules (Fig. since loops can be transformed to forward and backward oriented branches. cascaded signal paths
x/(1-xy) 3. 4: The three rules for simplifying signal flow charts. see Fig.

As before. the solution may sometimes be read directly from the diagram.a1
ρS
S11
ρL
Fig. When applied to S-matrices. But with the availability of powerful computer codes using the matrix formulations. Element S matrix b = ρia
a) Passive one-port
SFG
b = bq + ri a
b) Active one-port
7
. Applying the cascading rule and the parallel branch rule then yields
b1 ρL = S11 + S21 S12 a1 1 − S22 ρL
(3. the need to use the SFG has been reduced. 12]. As we have seen in this rather easy configuration. Then the signal path via b2 and a2 is added to S11. the SFG is a convenient tool for the analysis of simple circuits [8. first the loop consisting of S22 and ρL can be resolved. 5) and ask for b1/bs.4)
The same results would have been found applying Mason’s rule on this problem. ⎞ ⎟ ⎟ρ S ⎠
(3. For more complex networks there is a considerable risk that a signal path may be overlooked and the analysis soon becomes complicated. Finally one obtains
⎛ 1 ⎜ S 11 + S 21 S 12 ρ L ⎜ 1 − ρ L S 22 b1 ⎝ = bS ⎛ 1 1− ⎜ ⎜ S 11 + S 21 S 12 ρ L 1 − ρ S L 22 ⎝
⎞ ⎟ ⎟ ⎠ . yielding a loop with ρS. The SFG is also a useful way to gain insight into other networks. 5: A 2-port with a non-matched load The loop at port 2 involving S22 and ρL can be resolved. given a branch from b2 to a2 with the signal ρL *(1-ρL*S22).3)
As a more complicated example one may add a mismatch to the source (ρS = dashed line in Fig. such as feedback systems.

e.686 dB.
10
. 1 Np = 8.
•
Ideal isolator
•
⎛ 0 0⎞ S=⎜ ⎜ 1 0⎟ ⎟ ⎝ ⎠ The isolator allows transmission in one directly only. The values of the required resistors are k −1 R1 = Z 0 R1 R1 k +1
R2 = Z 0 2k 2 k −1
Port 1
R2
Port 2
where k is the voltage attenuation factor and Z0 the reference impedance. When port 3 connected to ground.•
Ideal. Ideal amplifier ⎛ 0 0⎞ S=⎜ ⎜G 0⎟ ⎟ ⎝ ⎠ with the gain G > 1. power dividers.g. with three resistors in a T circuit. It can be realized with three resistors in a triangle configuration. T junctions. The following three components have two of the above characteristics:
•
Resistive power divider: It consists of a resistor network and is reciprocal.g. to avoid reflections from a load back to the generator. reciprocal and matched at all three ports at the same time. An attenuator can be realized e. reciprocal attenuator
⎛ 0 e −α ⎞ ⎟ S=⎜ ⎜ e −α 0 ⎟ ⎝ ⎠ with the attenuation α in Neper.g. the resulting circuit is similar to a 2-port attenuator but not matched any more at port 1 and port 2. The attenuation in Decibel is given by A = -20*log10(S21). e.
3-ports Several types of 3-ports are in use. etc.g. matched at all ports but lossy.
⎛ 0 ⎜ S=⎜1 ⎜ 2 ⎜1 ⎝ 2 •
2 0 1 2
1
1 ⎞ 2⎟ 1 ⎟ 2⎟ 0 ⎟ ⎠
Port 1 Z0/3
Z0/3 Port 2 Z0/3 Port 3
The T splitter is reciprocal and lossless but not matched at all ports. It can be shown that a 3-port cannot be lossless. it is used e. circulators. 50 Ω.

8: 3-port circulator and 2-port isolator. A circulator. A signal entering the ideal circulator at one port is transmitted exclusively to the next port in the sense of the arrow (Fig. which lets power pass only from port 1 to port 2 (see section about 2-ports). This ferrite is normally magnetized into saturation by an external magnetic field. The μ+ and μrepresent the permeability seen by a right. 8). like the gyrator and other passive non-reciprocal elements contains a volume of ferrite.and left-hand circular polarized wave traversing the ferrite (Fig. The magnetic properties of a saturated RF ferrite have to be characterized by a μ-tensor. matched at all ports. They are strongly dependent on the bias field. but not reciprocal.Fig. 7: The two versions of the H10 waveguide T splitter: H-plane and E-plane splitter Using the losslessness condition and symmetry considerations one finds. the S matrix of the isolator has the following form:
⎛ 0 0 1⎞ ⎜ ⎟ S = ⎜ 1 0 0⎟ ⎜ 0 1 0⎟ ⎝ ⎠
When port 3 of the circulator is terminated with a matched load we get a two-port called isolator.
11
. 9). The real and imaginary part of each complex element μ are μ’ and μ’’.
2
Fig. for appropriate reference planes for H and E plane splitters
⎛1 1⎜ = ⎜ −1 2⎜ ⎜ 2 ⎝ −1 1 2 2⎞ ⎟ 2⎟ ⎟ 0 ⎟ ⎠ ⎛1 1⎜ S E = ⎜1 2⎜ ⎜ 2 ⎝ 1 1 − 2 2⎞ ⎟ − 2 ⎟ ⎟ 0⎟ ⎠
SH
The ideal circulator is lossless. Accordingly. The circulator can be converted into isolator by putting matched load to port 3.

the polarization of the waveguide mode is rotated counter clockwise by 45o by a
12
. 9 The real part μ’ (left) and imaginary part μ’’ (right) of the complex permeability μ. 12). power is only transmitted from port 1 to port 2. The magnetically polarized ferrite provides the required nonreciprocal properties.not | saturated | saturated
not saturated | saturated
low field losses
Fig. which has a vertically polarized H field in the waveguide on the left (Fig. 10 and 11 practical implementations of circulators are shown. The rightand left-hand circularly polarized waves in a microwave ferrite are μ+ and μ-. In Figs. As a result. After a transition to a circular waveguide. At the gyromagnetic resonance the right-hand polarized has high losses.
Fig.10 Waveguide circulator
ferrite disc port 1 port 3 ground planes
port 2
Fig. as can be seen from the peak in the right image. 11 Stripline circulator The Faraday rotation isolator uses the TE10 mode in a rectangular waveguide. A detailed discussion of the different working principles of circulators can be found in the literature [2.13]. from port 2 to port 3 and from port 3 to port 1.

In the waveguide on the left the backward wave arrives with a horizontal polarization. e. Finally. 12: Faraday rotation isolator The frequency range of ferrite-based. However. 13). non-reciprocal elements extends from about 50 MHz up to optical wavelengths (Faraday rotator) [13]. 4-port T-hybrids or magic tees.
attenuation foils
Fig. The S matrix of a 4-port As a first example let us consider a combination of E-plane and H-plane waveguide ‘T’s (Fig.g. a wave coming from the other side will have its polarization rotated by 45o clockwise as seen from the right side. while they hardly affect the forward wave. Therefore the Faraday isolator allows transmission only from port 1 to port 2. This configuration is called a Magic ‘T’ and has the S matrix:
⎛0 0 ⎜ 1 ⎜0 0 S= 1 1 2⎜ ⎜ ⎜1 −1 ⎝
1 1⎞ ⎟ 1 − 1⎟ 0 0⎟ ⎟ 0 0⎟ ⎠
13
. Then follows a transition to another rectangular waveguide which is rotated by 45o such that the rotated forward wave can pass unhindered. reciprocal elements.ferrite. The horizontal attenuation foils dampen this mode. it is shall be noted that all non-reciprocal elements can be made from a combination of an ideal gyrator (non-reciprocal phase shifter) and other passive.

Magic ‘T’. microstrip). (b. As usual the coefficients of the S matrix can be found by using the unitary condition and mechanical symmetries. A selection of possible waveguide couplers is depicted in Fig. In practice.
a
b
λg/4
c
d
e
Fig. This is shown in Fig. Contrary to 3-ports a 4-port may be lossless. Today. 13: Hybrid ‘T’. reciprocal and matched at all ports simultaneously. 180° hybrid. such as small matching stubs in the center of the ‘T’. Ideally there is no crosstalk between port 3 and port 4 nor between port 1 and port 2. respectively. 14: Waveguide directional couplers: (a) single-hole. coaxial line. Another important element is the directional coupler. The coupler is
14
. With a suitable choice of the reference planes the very simple S matrix given above results. which correspond to the Δ and Σ ports. Provided that both generators have equal amplitude and phase. certain measures are required to make out the ‘T’ a ‘magic’ one. Broadband versions of 180° hybrids may have a frequency range from a few MHz to some GHz. The bandwidth of a waveguide magic ‘T’ is around one octave or the equivalent H10-mode waveguide band. 14. The signal from generator 1 is split and fed with equal amplitudes into the E and H arm.Fig. the signals cancel at the Δ port and the sum signal shows up at the Σ port. They are widely used for signal combination or splitting in pickups and kickers for particle accelerators. and coupling is adjusted such that part of the power linked to a travelling wave in line 1 excites travelling waves in line 2. T-hybrids are often produced not in waveguide technology. The signal from generator 2 propagates in the same way. strip line. and the sum (Σ) and difference (Δ) signals are available from the H arm and E arm.e) multiple-hole types. 13 assuming two generators connected to the collinear arms of the magic T. There is a common principle of operation for all directional couplers: we have two transmission lines (waveguide.c) double-hole and (d. but as coaxial lines and printed circuits. In a simple vertical-loop pickup the signal outputs of the upper and lower electrodes are connected to arm 1 and arm 2.

A propagating wave in the main line excites electric and magnetic currents in the coupling hole. The directivity is the ratio of the desired coupled wave to the undesired (i. 6 dB. 14 is similar.directional when the coupled energy mainly propagates in a single travelling wave.g. two important figures are always required. The physical mechanism for the other couplers shown in Fig. and 20 dB with directivities usually better than 20 dB. Example: the 2-hole. i. e.e. Each coupling hole excites waves in both directions but the superposition of the waves coming from all coupling holes leads to a preference for a particular direction. at least two coupling mechanisms are necessary. Optimum directivity is only obtained in a narrow frequency range where the distance of the coupling holes is roughly λ/4. so they cancel. λ/4 coupler For a wave incident at port 1 two waves are excited at the positions of the coupling holes in line 2 (top of Fig.6)
. The holes need not be circular. The following relations hold for an ideal directional coupler with properly chosen reference planes S11 = S22 = S33 = S44 = 0 S21 = S12 = S43 = S34
15
(4. For the forward coupling the two waves add up in phase and all the power coupled to line 2 leaves at port 3. It can be shown that for α = 30° the electric and magnetic components cancel in one direction and add in the other and we have a directional coupler. 14). while the magnetic coupling is angle-dependent. In order to get directionality. i. etc. the coupling appears in the S matrix as the coefficient
⏐S13⏐=⏐S31⏐=⏐S42⏐=⏐S24⏐
with αc = -20 log⏐S13⏐in dB being the coupling attenuation. For larger bandwidths. Besides waveguide couplers there exists a family of printed circuit couplers (stripline. also known as a Bethe-coupler. For a backwards coupling towards port 4 these two wave have a phase shift of 180°. The electric coupling is independent of the angle α between the waveguides (also possible with two coaxial lines at an angle α). 10 dB. the coupling and the directivity. two coupling holes or electric and magnetic coupling. when there is no equal propagation in the two directions.
S α d = 20 log 31 S41
directivity [ dB] . microstrip) and also lumped element couplers (like transformers). 14b). Note that the ideal 3 dB coupler (like most directional couplers) often has a π/2 phase shift between the main line and the coupled line (90° hybrid). wrong direction) coupled wave.
Practical numbers for the coupling are 3 dB. takes advantage of the electric and magnetic polarizability of a small (d<<λ) coupling hole. they may be longitudinally or transversely orientated slots.e. To characterize directional couplers. For the Bethe coupler the electric coupling does not depend on the angle α between the waveguides. The single-hole coupler (Fig. For the elements shown in Fig. multiple hole couplers are used. crosses. 14. Each of these currents gives rise to travelling waves in both directions.e.

1
BASIC PROPERTIES OF STRIPLINES. the effect of the fringing fields may be described in terms of static capacities (see Fig. C’.
C tot = C p1 + C p 2 + 2C f 1 + 2C f 2
(5. 19: Characteristic impedance of striplines [14] For a mathematical treatment.2)
18
. The space around this conductor is filled with a homogeneous dielectric material.5. 5. With the static capacity per unit length. the static inductance per unit length. εr and the speed of light c the characteristic impedance Z0 of the line is given by
Z0 = ν ph = L′ C′ c εr
= 1 C ′c
1 L′C ′
(5. The total capacity is the sum of the principal and fringe capacities Cp and Cf. This line propagates a pure TEM mode. 20) [14]. L’. the relative permittivity of the dielectric.1)
Z0 = εr ⋅
Fig. MICROSTRIP AND SLOTLINES Striplines
A stripline is a flat conductor between a top and bottom ground plane.

5. It is thus an asymmetric open structure. Since there is a transversely inhomogeneous dielectric. the coupled wave leaves the coupler in the direction opposite to the incoming wave. 23 Microstripline: a) Mechanical construction. An exact field analysis for this line is rather complicated and there exist a considerable number of books and other publications on the subject [16.
metallic strip: ρ
ρ
Fig. and only part of its cross section is filled with a dielectric material.3 is also known as the Cohn nomographs [14].e. 23). the calculation of coupled lines and thus the design of couplers and related structures is also more
20
.4)
Z 0.even = Z 0
Z 0 = Z 0. In contrast to the 2-hole waveguide coupler this type couples backwards.
Z 0. i. An even simpler way to make such devices is to use a section of shielded 2-wire cable.even
where C0 is the voltage coupling ratio of the λ/4 coupler. The stripline coupler technology is rather widespread by now. 24). and very cheap high quality elements are available in a wide frequency range. Due to the dispersion of the microstrip. 16) very simple design formulae can be given.2 Microstrip
A microstripline may be visualized as a stripline with the top cover and the top dielectric layer taken away (Fig. b) Static field approximation [16].odd Z 0. 17]. This has several implications such as a frequency-dependent characteristic impedance and a considerable dispersion (Fig. only a quasi-TEM wave exists. For a quarter-wave directional coupler (single section in Fig.odd = Z 0
1 + C0 1 − C0 1 − C0 1 + C0
(5.A graphical presentation of Equations 5.

and easy access to the surface for the integration of active elements. With all the disadvantages mentioned above in mind. Microstrip circuits are also known as Microwave Integrated Circuits (MICs).
εr. 24: Characteristic impedance (current/power definition) and effective permittivity of a microstrip line [16]
εr
εr
εr2 εr1 εr εr εr εr εr2 εr1 εr2
εr1
Fig. In Figs. once a conductor pattern has been defined. 26: Various transmission lines derived from MIC microstrip [17]
21
. The mains reasons are the cheap production. Microstrips tend to radiate at all kind of discontinuities such as bends. one may question why they are used at all. 25 and 26 various planar printed transmission lines are depicted. 25: Planar transmission lines used in Fig.eff
Fig. The microstrip with overlay is relevant for MMICs and the strip dielectric wave guide is a ‘printed optical fibre’ for millimeter-waves and integrated optics [17]. changes in width. A further technological step is the MMIC (Monolithic Microwave Integrated Circuit) where active and passive elements are integrated on the same semiconductor substrate.complicated than in the case of the stripline. through holes etc.

A unique feature of the slotline is that it may be combined with microstrip lines on the same substrate.5. This. The characteristic impedance and the effective dielectric constant exhibit similar dispersion properties to those of the microstrip line. in conjunction with through holes. b) Field pattern (TE approximation). It is essentially a slot in the metallization of a dielectric substrate as shown in Fig.3
Slotlines
The slotline may be considered as the dual structure of the microstrip. permits interesting topologies such as pulse inverters in sampling heads (e. for sampling scopes). d) Magnetic line current model. Assuming the upper microstrip to be the input. Fig.g. c) Longitudinal and transverse current densities. Reproduced from [16] with permission of the author. 27. e. the signal leaving the circuit on the lower microstrip is inverted since this microstrip ends on the opposite side of the slotline compared to the input.
Fig. Printed slotlines are also used for broadband pickups in the GHz range. 28 shows a broadband (decade bandwidth) pulse inverter. for stochastic cooling [15].
22
.g. 27: Slotlines a) Mechanical construction.

.4)
However.1. for practical applications the bandwidth of a single λ/4 transformer is often not sufficient. Such a spiral can be easily visualized by connecting an open or shorted transmission line to a network analyser and displaying the reflection as a function of frequency in the Smith Chart. or in other words it reflects is about the origin of the SC. i. the wave runs along line twice. the transformation over a lossy line has to be considered.Exercise: Imagine a Cartesian coordinate system superimposed and visualize a few important reference points. parallel elements are added in an SC normalized to Y0. Since in practise there are no lossless lines. Adding a lossless line of with a length l to a given impedance makes ρ turn clockwise about the center of the SC by 4π x l/λ rad.
25
. Z6
Impedances are added in an SC normalized to Z0 when they are connected in series. A short circuit is transformed into an open circuit and vice versa.j25 Ω Z0 = 50 Ω Z0 = 50 Ω Z0 = 50 Ω Z0 = 100 Ω Z0 = 50 Ω Z0 = 100 Ω
Try also parallel and serial combinations of Z1 . It leads to a logarithmic spiral which is not easy to draw but can be constructed pointwise. Note that a line of length λ gives two full rotations.
ρ=0 ρ = -1 ρ = +1 ρ = +j ρ = -j
R/Z0 = 1 R/Z0 = 0 R/Z0 = ∞ R/Z0 = 0 R/Z0 = 0
X/Z0 = 0 X/Z0 = 0 X/Z0 = ∞ X/Z0 = +1 X/Z0 = 1
Matched load Short Open Shorted λ/8 line (Z0) Shorted 3 λ/8 line (Z0)
The loci of constant real and imaginary part are obtained from the conformal mapping of Eq. Thus multistage line transformers are used which requires a renormalization of the Smith Chart for each line impedance. Admittances. Such a line is often used as a “λ/4 transformer” and with line termination Z1 and a line input impedance Z2 we obtain
2 Z1Z 2 = Z 0
(6. π/2 for λ/8. As a further exercise one may read modules and phase of ρ for a few impedances: Z1 = 25 Ω Z2 = 100 Ω Z3 = 50 + j50 Ω Z4 = 50 + j50 Ω Z5 = + j200 Ω Z6 = 100 . π for λ/4. which is due to the fact that since the reflection coefficient is plotted. etc. Thus a line of length λ/4 turns ρ by π rad. 6.. i.e.e.

31). Condition: |S11| → min. On a vector network analyzer with Smith Chart display option the locus of S11 will be a circle (dashed thick red circle in Fig. The quality factor of the resonant circuit together with the external circuitry is called the loaded Q factor QL and Qext is the Q factor of the external circuitry. or by directly searching the points on the network analyser using special marker settings.e.
•
f0 gives the center frequency of the resonator. as described below. such as a closed cavity. In the following. In practise one has to connect the external measurement apparatus to the resonant circuit. The seven frequency points f0 through f6 can be used to calculate the different Q values:
Q0 = f 0 / ( f5 − f6 ) QL = f 0 / ( f1 − f 2 ) Qext = f 0 / ( f3 − f 4 )
The points f0 through f6 can be found geometrically by drawing the blue artificial circles and lines [6. that is. i. Usually one wants to determine the unloaded quality factor Q0 of an unperturbed system. The three Q factors are related by
1 1 1 = + Q L Q0 Q ext
(6.7]. since this results in a distortion.
Q=
ωW
P
where ω is the angular frequency. which will change the initial conditions. we consider the circle to be in the “detuned short” position.Measurement of the quality factor in the Smith Chart The quality factor of a resonant circuit is defined as the ratio of the stored energy W over the energy dissipated P in one cycle. it is symmetric about the horizontal axis and for frequency far off-resonance it approaches the short position ρ = 1. the reflection coefficient as a function of frequency. The electrical delay should not be used for rotating the circle.5)
The Q factor of a resonance peak or dip can be calculated from the center frequency f0 and the 3 dB bandwidth Δf as
Q=
f0 Δf
A very convenient way of characterizing resonant circuits is by measuring the locus of S11. In practise. The size of this circle increases with the coupling to the resonant circuit. the circle displayed on the Smith Chart can be easily rotated into this position by adjusting the phase offset. Procedure in Smith Chart: o Resonator in “detuned short” position
26
. It may be rotated about the origin of the Smith Chart due to transmission lines between the network analyser and the resonant circuit.

QL = Q0/2. Condition: Z = ±j in detuned open position. The phase swing is 180°. 194-202. At resonance all the available generator power is coupled to the resonance circuit. o o o Resonator in “detuned open” position Marker format: Z Search for the two points where Z = ±j ⇒ f3 and f4
There are three ranges of the coupling factor β defined by
β=
Q0 Qext
(6. The phase swing is larger than 180°. Vol. The locus of ρ in the detuned short position is left of the center of the SC. which is equivalent to Y = ±j in detuned short position.6)
or. The locus of ρ touches the center of the SC. IEEE-T-MTT. the phase of the complex reflection factor ρ is required to make the distinction.o o o
Resonator in “detuned short” position Marker format: Re{S11} + jIm{S11} Search for the two points where |Im{S11}| → max ⇒ f1 and f2
•
The external QE can be calculated from f3 and f4. 13 No 2. The phase swing is smaller than 180°.6
QL =
This allows us to define:
Q0 1+ β
(6. REFERENCES [1] K. Kurokawa. ACKNOWLEDGEMENTS The authors would like to thank Stuart Eyres. Andrew Collins and June Prince from STFC Daresbury Laboratory for their hard work in reproducing this publication.or undercritical.
28
. March 1965. 6.6)
•
Critical Coupling: β = 1. using Eq.
•
•
When using a network analyzer with a Cartesian display for ⏐ρ⏐ one cannot decide whether the coupling is over. Power waves and the scattering matrix. Undercritical Coupling: (0 < β < 1). Overcritical coupling: (1 < β < ∞). The center of the SC is inside the locus of ρ.

10)
Similarly to the S matrix.8)
So far. one would like to return to S parameters
T T T S11 = 12 .11)
With the direction of I2 chosen in Fig. the all other ports have to be open.7)
T11T22 − T12T21 = 1
(AI. respectively. the dependent variables U1 and U2 are written as a Z matrix:
U1 = Z11I1 + Z12 I 2 U 2 = Z 21I1 + Z 22 I 2
or (U ) = ( Z ) ⋅ ( I )
(AI. 1)
⎛ U1 ⎞ ⎛ A B ⎞ ⎛ U 2 ⎞ ⎛ A11 A12 ⎞ ⎛ U 2 ⎞ ⎜ ⎟=⎜ ⎟=⎜ ⎟⎜ ⎟ ⎟⎜ ⎝ I1 ⎠ ⎝ C D ⎠ ⎝ − I 2 ⎠ ⎝ A21 A22 ⎠ ⎝ − I 2 ⎠
(AI.and Y-matrices are not easy to apply for cascaded 4-poles (2-ports).In practice. In an analogous manner. Considering the current I1 and I2 as independent variables. in contrast to the S parameter measurement. where matched loads are required. the so-called ABCD matrix (or A matrix) has been introduced as a suitable cascaded network description in terms of voltages and currents (Fig. 1 a minus sign appears for I2 of a first 4-pole becomes I1 in the next one. A description in voltages and currents is also useful in many cases. S12 = T11 − 12 21 T22 T22 T 1 S21 = . It can be shown that the ABCD matrix of two or more cascaded 4-poles becomes the matrix product of the individual ABCD-matrices [3]
31
. after having carried out the T matrix multiplication. S12 = − 21 T22 T22
For a reciprocal network (Sij = Sji) the T-parameters have to meet the condition det T = 1
(AI.9)
where Z11 and Z22 are the input and output impedance. a Y matrix (admittance matrix) can be defined as
I1 = Y11U1 + Y12U 2 I 2 = Y21U1 + Y22U 2
or ( I ) = (Y ) ⋅ (U )
(AI. When measuring Z11. the Z. Thus. we have been discussing the properties of the 2-port mainly in terms of incident and reflected waves a and b.

going from N1 to N2. and N2 is linked to all signals previously leaving from N. Each node signal represents the sum of the signals carried by
all branches entering it. 2. 7. 5. A first-order loop is the product of branch transmissions. 9. each representing one S parameter and indicating direction. When there are no direct signal loops. The transmission coefficients of parallel signal paths are to be added.1)
where M and M’ are square matrices with n rows and columns. x represents the n independent variables (sources) and y the n dependent variables. The elements of M and M’ appear as transmission coefficients of the signal path. 10. AII. 6. The SFG has a number of points (nodes) each representing a single wave ai or bi. 8. Nodes are connected by branches (arrows). All other nodes are dependent signal nodes.2)
The purpose of the SFG is to visualize physical relations and to provide a solution algorithm of Eq. A node may be the beginning or the end of a branch (arrow). as is generally the case in practise. which is equivalent to the usual S parameter definition b = Sa (AII. N1 has all signals (branches) previously entering N. the previous equation simplifies to y = Mx. 4. 3. A second-order loop is the product of two non-touching first-order loops. An elementary loop with the transmission coefficient S beginning and ending at a node N may be replaced by a branch (1-S)-1 between two nodes N1 and N2.2 by applying a few rather simple rules: 1.
35
. Nodes showing no branches pointing towards them are source nodes. The transmission coefficients of cascaded signal paths are to be multiplied. An SFG is feedback-loop free if a numbering of all nodes can be found such that every branch points from a node of lower number towards one of higher number. and an nth-order loop is the product of any n non-touching first-order loops. starting from a node and going along the arrows back to that node without touching the same node more than once.Appendix II The SFG is a graphical representation of a system of linear equations having the general form: y = Mx + M’y (AII.